The number of fish in a tank decreases by x% each year. Given that the number of fish halves in 8 years, work out the value of x. Give your answer to 1dp.
(3marks)
I've tried this question but am not sure of my answer. Could someone please go through it for me, thanks!
Certainly, if you would post your attempt!
Sure, so I used the formula Final amount=original amount* (1-x/100)^time.
I rearranged the equation to make x the subject:
X=time√final amount/original amount -1*100
Then I subbed in values:
X=8√50/100 -1 *100
(I used 50 and 100 because 100 would half to a final amount of 50)
I worked out x to be -8.3 to 1dp.
I don't think it should b negative I don't know where I went wrong.
well ok, then your way
so let
(1 - x/100) = z
1/2 = z^8
ln .5 = 8 ln z
so
ln z = -.693/8 = - .0866
then z = e^-.0866 = .917
x/100 = 1-.917 = .083
so
x = 8.3 %
note my way uses continuous rate, yours is yearly change.
The number of fish in a tank decreases by x% each year. Given that the number of fish halves in 8 years,
(1-x/100)^t = (1/2)^(t/8)
t log(1-x/100) = t/8 log(1/2)
log(1-x/100) = log(2^(-1/8))
1 - x/100 = 2^(-1/8)
x/100 = 1 - 2^(-1/8)
x = 100(1 - 2^(-1/8)) = 8.3
To solve this problem, we need to understand the relationship between the decrease in the number of fish and the passing of years.
Let's start by defining the variables:
- N₀: initial number of fish
- Nₓ: number of fish after x years
We are given that the number of fish halves in 8 years. This means that after 8 years, the number of fish will be N₀/2.
Now, we can set up the equation to represent the decrease in the number of fish each year. Since the decrease per year is given as a percentage, we can express it as a decimal fraction by dividing it by 100.
Nₓ = N₀ * (1 - x/100)^8
We know that after 8 years, the number of fish halves:
N₈ = N₀/2
Substituting these values into the equation, we have:
N₀/2 = N₀ * (1 - x/100)^8
Now, we can solve for x.
Divide both sides of the equation by N₀:
1/2 = (1 - x/100)^8
Take the eighth root of both sides:
(1/2)^(1/8) = 1 - x/100
Now, rearrange the equation to solve for x:
x/100 = 1 - (1/2)^(1/8)
Multiply both sides by 100:
x = 100 - 100 * (1/2)^(1/8)
Using a calculator, we can evaluate the right side of the equation to find the value of x. Rounding to 1 decimal place, we get:
x = 100 - 100 * (1/2)^(1/8) ≈ 9.5
Therefore, the value of x is approximately 9.5%.
dn/dt = -(x/100) n
dn/n = -(x/100) dt
ln n = (-x/100 + C
n = e^(-x/100 + C)t
or
n = c e^-(x/100)t
at t = 0, n = N so c = N
at t = 8, n = N/2
so
c = N
and
N/2 = N e^-8x/100
or
.5 = e^-.08x
ln .5 = -.08 x
-.693 = -.08 x
x = 8.725 %